 So, we just developed a general approach to finding the determinant where we could expand along any row or column, multiply the row or column entry by the minor, and then add or subtract the products of the minors with the entries. But the problem we ran into was how can we decide when to add or when to subtract? So, let's think about that. Our theorem relies having our determinant in a particular form where the ij entry is in the first row, first column position. So, to move the ij entry to the first row, first column position, so we can apply our theorem, we'll need to do the following. First, we'll need to switch rows i minus one times, and that'll bring our entry to the top. And next, we'll need to switch columns j minus one times, and that'll bring our entry to the first column. Now, every time we switch a row or switch a column, we'll change the signs. So, this means we'll switch the signs i minus one plus j minus one, or i plus j minus two times, and that means we'll multiply the minor by negative one to power i plus j minus two. And we can simplify that a little bit to minus one to power i plus j. And this leads to the following definition. The ijth cofactor of a square matrix M is negative one to power i plus j times the minor of i and j. So, putting all of these pieces does give us a general approach, really, to finding the determinant. So, our general strategy is going to be the following. We'll choose any row or column to expand along. We'll find the cofactors. We'll multiply the cofactors by the row or column entries, and then add. Okay, so let's put all of these things together to find the determinant of a three by three matrix. We'll put on our procedure for reference. Now, since we can choose any row or column to expand on, we might worry that we have to choose the right row to get the determinant. Well, let's see if it makes a difference. So, we'll expand along the first row and compare that to what we get by expanding along the, oh, I don't know, how about the third column? So, if we use the first row, we'll take the first row first column entry two times the minor, which is the sub-matrix we get by eliminating the first row and first column. And since this is the value from the first row first column, we'll multiply this by minus one to the power one plus one. Our next entry is seven. We'll form the minor by eliminating the row and column to get the sub-matrix. And since this is the first row second column entry, we'll multiply this by negative one to power one plus two. And finally, the third entry, three, is going to be multiplied by the minor and then multiplied by negative one to power one plus three. And if we can evaluate these powers, negative one to the power one plus one, well, since that's an even power, that quantity is just going to be one. Negative one to the power one plus two, that's an odd power, so that means we'll subtract the second term. Negative one to the power one plus three, that's an even power, so it's the same as just adding this term, finding our determinants, and adding or subtracting as appropriate gives us our determinant, negative 86. And again, because we found the determinant by expanding along any row or column, maybe we expanded along the wrong one and got the wrong value. Well, let's check it out by expanding along the third column. So let's take a look at that. We'll have three times negative one to power one plus three times the minor, plus five times negative one to power two plus three times the minor, plus six times power negative one to power three plus three times the minor. And we'll find those determinants, and when we add everything together, we get a determinant of negative 86. And so notice that we found this determinant to be negative 86 when we expanded along the first row, but we got the same result expanding along the third column. And in fact, this is no coincidence, this is an example of a general theorem that says the determinant of a matrix does not depend on the row or column used to expand.